Interval Notation Calculator

Interval notation is a compact way to describe every real number between two endpoints. This calculator reads a single interval and translates it into an equivalent inequality involving x. An interval begins with a parenthesis or square bracket, lists its lower and upper endpoints separated by a comma, and ends with a parenthesis or square bracket. The endpoints must be written from least to greatest. A square bracket means that a finite endpoint belongs to the interval. For example, [2, 6] includes both 2 and 6 and translates to 2 less than or equal to x less than or equal to 6. A parenthesis means that the endpoint is excluded. Therefore, (2, 6) contains values strictly between 2 and 6 but contains neither boundary. Mixed intervals such as (-2, 4] exclude the lower endpoint while including the upper endpoint. Infinity symbols indicate that an interval continues forever in one direction. Since infinity is not a real number that can be reached or included, an infinite endpoint always uses a parenthesis. The interval [3, infinity) means x is greater than or equal to 3. The interval (negative infinity, 7) means x is less than 7. Writing a bracket beside infinity would incorrectly imply that infinity is a member of the real-number set. Intervals appear throughout algebra and calculus. They describe domains and ranges, solution sets, increasing and decreasing regions, continuity, confidence intervals, and valid parameter values. Compared with a long inequality, the notation makes endpoint inclusion immediately visible. It also combines naturally with the union symbol when a solution has disconnected pieces, although this calculator focuses on one continuous interval at a time. A number-line graph provides a useful visual check. Draw an open circle for a parenthesis and a filled circle for a bracket, then shade all values between the endpoints. For an infinite interval, draw an arrow in the continuing direction. The inequality produced by the calculator should describe exactly the same circles and shading. A finite left endpoint compares to x with a less-than sign, while a finite right endpoint compares from x toward the boundary. The order of symbols matters. Parentheses can serve both as interval delimiters and as ordinary grouping marks in other mathematical contexts, so a comma and two endpoints are necessary to identify an interval. Square brackets may also appear in matrices or lists, but the two-endpoint form makes their interval meaning clear here. Decimal and negative endpoints are accepted as ordinary real-number boundaries. To verify an interpretation, test one point in the interior and then test each finite endpoint. Every interior point should satisfy the resulting inequality. An endpoint should satisfy it only when the corresponding interval mark is a bracket. This method quickly identifies reversed signs, incorrectly included boundaries, and endpoints entered in the wrong order.